The cap product equality conjecture for right-angled Artin groups

Let $A_\Gamma$ be a right-angled Artin group associated to a finite simple graph $\Gamma$. For a second homology class $\alpha\in H_2(A_\Gamma)$, let $M_\alpha$ denote the associated skew-symmetric bilinear form, and let $\operatorname{gen}(\alpha)$ be the minimal genus of a surface representing $\alpha$. The cap product inequality is the inequality relating $\operatorname{gen}(\alpha)$ to the cap bound $\frac{1}{2}\operatorname{rank}(M_\alpha)$. Cap product equality conjecture. The cap product inequality is an equality for any right-angled Artin group. This would answer the paper's question about whether the cap bound determines the minimal genus in full generality; the preceding results establish equality in several classes of examples, but the general case remains open.